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Abstract:

A prediction of whether a point on a computer-generated surface is
adjacent to laminar or turbulent flow is made using a transition
prediction technique. A plurality of instability modes are obtained, each
defined by one or more mode parameters. A vector of regressor weights is
obtained for the known instability growth rates in a training dataset.
For an instability mode in the plurality of instability modes, a
covariance vector is determined. A predicted local instabilty growth rate
at the point is determined using the covariance vector and the vector of
regressor weights. Based on the predicted local instability growth rate,
an n-factor envelope at the point is determined.

Claims:

1. A computer-implemented method for predicting whether a point on a
computer-generated surface is adjacent to laminar or turbulent fluid
flow, the method comprising: obtaining, using a computer, a plurality of
instability modes, wherein one or more mode parameters define each
instability mode; obtaining, using the computer, a vector of regressor
weights for a set of known instability growth rates in a training
dataset; for an instability mode in the plurality of instability modes:
determining, using the computer, a covariance vector comprising a
covariance of a predicted local instability growth rate for the point
with respect to the set of known instability growth rates in the training
dataset; and determining, using the computer, a predicted local growth
rate at the point for the instability mode using the vector of regressor
weights and the covariance vector; and determining, using the computer,
an n-factor envelope at the point using the predicted local instability
growth rate, wherein the n-factor envelope is indicative of whether the
point is adjacent to laminar or turbulent flow.

2. The computer-implemented method of claim 1, further comprising:
determining whether the fluid flow at the point is turbulent or laminar
based on whether the n-factor envelope at the point exceeds a threshold
value, wherein if the n-factor envelope at the point exceeds the
threshold value, then the point is adjacent to turbulent flow and wherein
if the n-factor envelope at the point is less than the threshold value,
then the point is adjacent to laminar flow.

3. The computer-implemented method of claim 1, further comprising:
obtaining, using the computer, a plurality of boundary-layer properties
at the point on the computer-generated surface using a steady-state
solution of a fluid flow in a region adjacent to the point, wherein the
training dataset includes a training input vector associated with each
known instability growth rate, wherein each training input vector
includes training boundary-layer properties and at least one training
instability mode parameter, wherein the vector of regressor weights is
based on a covariance matrix, wherein the covariance matrix has elements
that are covariances of one known instability mode with respect to
another known instability mode, wherein a covariance of a first known
instability mode with respect to a second known instability mode is based
on a distance between a first and a second training input vectors
associated with a first and a second known instability growth rates,
respectively, wherein the predicted local instability growth rate is
associated with the plurality of boundary-layer properties and at least
one training instability mode parameter describing an instability mode,
and wherein the covariance for the predicted local instability growth
rate with respect to a known instability growth rate is based on the
distance from the plurality of boundary-layer properties and the at least
one training instability mode parameter to the training input vector
associated with the known instability growth rate.

4. The computer-implemented method of claim 3, further comprising:
determining whether the fluid flow at the point is turbulent or laminar
based on whether the n-factor envelope at the point exceeds a threshold
value, wherein if the n-factor envelope at the point exceeds the
threshold value, then the point is adjacent to turbulent flow and wherein
if the n-factor envelope at the point is less than the threshold value,
then the point is adjacent to laminar flow.

5. The computer-implemented method of claim 3, wherein the training
dataset is a subset of one partition of a plurality of partitions of a
larger dataset, wherein the known instability growth rates are added to
the subset from the partition based on a prediction error associated with
predicting the local instability growth rate, and wherein the one
partition is chosen based on the boundary-layer properties or the at
least one training instability mode parameter.

6. The computer-implemented method of claim 1, wherein the plurality of
instability modes are of the stationary crossflow type, and wherein each
of the plurality of instability modes has a temporal frequency of zero.

7. (canceled)

8. The computer-implemented method of claim 3, wherein the covariance
matrix is based on a squared exponential covariance function.

9. The computer-implemented method of claim 3, wherein the covariance
matrix and the covariance vector are based on a squared exponential
covariance function.

11. The computer implemented method of claim 1, wherein the training
dataset is constructed by: obtaining a larger dataset of known
instability growth rates; adding a subset of known instability growth
rates that are in the larger dataset of known instability growth rates to
the training dataset; determining a prediction error between a known
instability growth rate in the larger dataset that is not in the training
dataset and the predicted local instability growth rate; and based on the
prediction error, adding the known instability growth rate to the
training dataset from the larger dataset.

12. (canceled)

13. (canceled)

14. The computer implemented method of claim, 1 further comprising:
determining a confidence measure for the predicted local instability
growth rate, wherein the confidence measure is based on the covariance of
the predicted local instability growth rate with respect to the known
instability growth rates in the training dataset.

15. The computer implemented method of claim 14, further comprising: if
the confidence measure indicates error above a threshold, adding
additional known instability growth rates to the training dataset.

16. (canceled)

17. A nontransitory computer-readable medium storing computer-readable
instructions which, when executed on a computer, perform a method for
predicting whether a point on a computer-generated surface is adjacent to
laminar or turbulent fluid flow, the medium including instructions for:
determining a covariance vector comprising a covariance of a predicted
local growth rate for the point with respect to a set of known
instability growth rates in a training dataset; determining the predicted
local instability growth rate at the point for an instability mode using
a vector of regressor weights and the covariance vector, and wherein, the
vector of regressor weights correspond to the set of known instability
growth rates in the training set; and determining an n-factor envelope at
the point using the predicted local instability growth rate, wherein the
n-factor envelope is indicative of whether the point is adjacent to
laminar or turbulent flow.

18. The computer-readable medium of claim 17, further comprising
instructions for: determining whether the fluid flow at the point is
turbulent or laminar based on whether the n-factor envelope at the point
exceeds a threshold value, wherein if the n-factor envelope at the point
exceeds the threshold value, then the point is adjacent to turbulent flow
and wherein if the n-factor envelope at the point is less than the
threshold value, then the point is adjacent to laminar flow.

19. The computer-readable medium of claim 17, the instructions further
comprising: obtaining, using the computer, a plurality of boundary-layer
properties at the point on the computer-generated surface using a
steady-state solution of a fluid flow in a region adjacent to the point,
wherein the training dataset includes a training input vector associated
with each known instability growth rate, wherein each training input
vector includes training boundary-layer properties and at least one
training instability mode parameter, wherein the vector of regressor
weights is based on a covariance matrix, wherein the covariance matrix
has elements that are covariances of one known instability mode with
respect to another known instability mode, wherein the covariance of a
first known instability mode with respect to a second known instability
mode is based on a distance between a first and a second training input
vectors associated with a first and a second known instability growth
rates, respectively, wherein the predicted local growth rate is
associated with the plurality of boundary-layer properties and at least
one training instability mode parameter describing an instability mode,
and wherein the covariance for the predicted local instability growth
rate with respect to a known instability growth rate is based on a
distance from the plurality of boundary-layer properties and at least one
training mode parameter to the training input vector associated with the
known instability growth rate.

20. The computer-readable medium of claim 19, further comprising
instructions for: determining whether the fluid flow at the point is
turbulent or laminar based on whether the n-factor envelope at the point
exceeds a threshold value, wherein if the n-factor envelope at the point
exceeds the threshold value, then the point is adjacent to turbulent flow
and wherein if the n-factor envelope at the point is less than the
threshold value, then the point is adjacent to laminar flow.

21. The computer-readable medium of claim 19, wherein the training
dataset is a subset of one partition of a plurality of partitions of a
larger dataset, wherein the known instability growth rates are added to
the subset from the partition based on a prediction error associated with
predicting the local instability growth rate, and wherein the one
partition is chosen based on the boundary-layer properties or the at
least one training instability mode parameter.

22. The computer-readable medium of claim 17, wherein the plurality of
instability modes is of the stationary crossflow type, and wherein each
of the plurality of instability modes has a temporal frequency of zero.

23. (canceled)

24. The computer-readable medium of claim 17, wherein the covariance
matrix is based on a squared exponential covariance function.

25. The computer-readable medium of claim 17, wherein the covariance
matrix and the covariance vector are based on a squared exponential
covariance function.

27. The computer-readable medium of claim 17, wherein the training
dataset is constructed by: obtaining a larger dataset of known
instability growth rates; adding a subset of known instability growth
rates that are in the larger dataset to the training dataset; determining
a prediction error between a known instability growth rate in the larger
dataset that is not in the training dataset and the predicted local
instability growth rate; and based on the prediction error, adding the
known instability growth rate to the training dataset from the larger
dataset.

28. (canceled)

29. (canceled)

30. The computer-readable medium of claim 17 further comprising
instructions for: determining a confidence measure for the predicted
local instability growth rate, wherein the confidence measure is based on
the covariance of the predicted local instability growth rate with
respect to the known instability growth rates in the training dataset.

31. The computer-readable medium of claim 30 further comprising
instructions for: if the confidence measure indicates error above a
threshold, adding additionally known instability growth rates to the
training dataset.

32. (canceled)

33. A computer system for predicting whether a point on a
computer-generated surface is adjacent to laminar or turbulent fluid
flow, the system comprising: a computer memory, a computer processor for
executing computer-readable instructions, the instructions configured to
cause the computer processor to: obtain a plurality of instability modes,
wherein one or more mode parameters define each instability mode; obtain
a vector of regressor weights for a set of known instability growth rates
in a training dataset; for an instability mode in the plurality of
instability modes: determine a covariance vector comprising a covariance
of a predicted local instability growth rate for the point with respect
to the set of known instability growth rates in the training dataset; and
determine the predicted local instability growth rate at the point for
the instability mode using the vector of regressor weights and the
covariance vector; and determine an n-factor envelope at the point using
the predicted local instability growth rate, wherein the n-factor
envelope is indicative of whether the point is adjacent to laminar or
turbulent flow.

Description:

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] The present application is a continuation of U.S. application Ser.
No. 13/069,374, filed Mar. 22, 2011, which is incorporated herein by
reference in its entirety.

BACKGROUND

[0003] 1. Field

[0004] This application relates generally to simulating a fluid flow over
a computer-generated surface and, more specifically, to predicting
whether a point on the surface is adjacent to laminar or turbulent flow.

[0005] 2. Description of the Related Art

[0006] Aerodynamic analysis of an aircraft moving through a fluid
typically requires an accurate prediction of the properties of the fluid
surrounding the aircraft. Accurate aerodynamic analysis is particularly
important when designing aircraft surfaces, such as the surface of a wing
or control surface. Typically, the outer surface of a portion of the
aircraft, such as the surface of a wing, is modeled, either physically or
by computer model, so that a simulation of the fluid flow can be
performed and properties of the simulated fluid flow can be measured.
Fluid-flow properties are used to predict the characteristics of the
wing, including lift, drag, boundary-layer velocity profiles, and
pressure distribution. The flow properties may also be used to map
laminar and turbulent flow regions near the surface of the wing and to
predict the formation of shock waves in transonic and supersonic flow.

[0007] A computer-generated simulation can be performed on a
computer-generated aircraft surface to simulate the fluid dynamics of a
surrounding fluid flow. The geometry of the computer-generated aircraft
surface is relatively easy to change and allows for optimization through
design iteration or analysis of multiple design alternatives. A
computer-generated simulation can also be used to study situations that
may be difficult to reproduce using a physical model, such as supersonic
flight conditions. A computer-generated simulation also allows a designer
to measure or predict fluid-flow properties at virtually any point in the
model by direct query, without the difficulties associated with physical
instrumentation or data acquisition techniques. In this way,
computer-generated simulations allow a designer to select an aircraft
surface design that optimizes particular fluid-flow characteristics.

[0008] In some cases, a portion of an aircraft surface, such as a wing
surface, can be optimized to maximize regions of laminar flow. A region
of fluid flow may be considered laminar when the flow tends to exhibit a
layered or sheet-like flow. In laminar-flow regions there is little
mixing between the layers or sheets of fluid flow having different fluid
velocities. Laminar flow can be contrasted to turbulent flow, which tends
to exhibit chaotic or erratic flow characteristics. In turbulent-flow
regions there is a significant amount of mixing between portions of the
fluid flow having different fluid velocities.

[0009] Near the surface of a wing, the fluid flow typically begins as
laminar flow at the leading edge of the wing and becomes turbulent as the
flow progresses to the trailing edge of the wing. The location on the
surface of the wing where the fluid flow transitions from laminar to
turbulent is called a transition point. The further the transition point
is from the leading edge, the larger the region of laminar flow.

[0010] There are many advantages to aircraft utilizing laminar flow over
large portions of the fuselage and wing surfaces. In general, laminar
flow dissipates less energy than turbulent flow. Increasing the
proportion of laminar flow regions over a wing surface reduces drag, and
therefore, reduces fuel burn, emissions, and operating costs.

[0011] According to one model, the transition to turbulent flow is caused
by the growth of instabilities in the boundary-layer fluid flow adjacent
to the aircraft surface. These instabilities may be initiated by, for
example, surface contamination, roughness, vibrations, acoustic
disturbances, shockwaves, or turbulence in the free-stream flow. The
instabilities start out as small, periodic perturbations to the fluid
flow near the aircraft surface, then grow or decay depending on the
properties of the boundary layer, such as flow velocity and temperature
profiles. At first, when the instabilities are small, their behavior is
similar to sinusoidal plane wave instabilities and can be described by
linearized perturbation equations. As the unstable modes grow in
amplitude, nonlinear interactions become dominant. Following the
nonlinear growth, the laminar instabilities begin causing intermittent
spots of turbulence, which spread and eventually merge together,
resulting in a fully turbulent boundary layer.

[0012] When predicting the location where a laminar flow transitions to
turbulent flow, designers may consider many different types of
instabilities. These types include Tollmien-Schlichting (TS) wave
instabilities and crossflow vortices. The type of instability may depend,
in part, on the geometry of the aircraft, such as the degree of sweep of
the wing.

[0013] For a given instability type (e.g., TS wave or crossflow vortex),
there are typically multiple individual instability modes that may be
defined using mode parameters, such as temporal frequency and/or spatial
spanwise wave number. By considering a range of individual instability
modes when simulating a fluid flow around the aircraft surface, designers
may account for a variety of potential instability sources.

[0014] In general, transition prediction techniques allow a designer to
estimate the point on an aircraft surface where laminar flow first
transitions to turbulent flow. In some cases, designers may attempt to
maximize regions of laminar flow by designing the surface of the wing so
that the transition point is as far from the leading edge as possible.
Producing useful results often requires running complex simulations over
a wide range of design variables and flight conditions. Unless the
transition prediction technique is efficient and easy to use, running
multiple complex simulations may be prohibitively time-consuming in the
earlier stages of aircraft design where major configuration changes are
likely.

[0015] One transition prediction technique is based on linear stability
theory (LST), which may be used to model the growth of instabilities in a
boundary-layer fluid flow around a computer-generated aircraft surface.
LST models these instabilities as spatio-temporal waves that are
amplified or attenuated as the flow progresses along the boundary layer.
This modeling requires the solution of an eigenvalue problem. Input to an
LST-based analysis includes a boundary-layer solution and values
parameterizing a selected instability mode (e.g., wave number and
frequency). The input boundary-layer solution includes, for example,
boundary-layer properties such as flow velocity and temperature and can
be determined using a time-invariant computational fluid dynamics (CFD)
simulation module. As an output, LST-based analysis computes a local
instability growth rate associated with the selected instability mode at
a given point on the aircraft surface.

[0016] While LST-based analysis may produce accurate results, LST-based
analysis may be prohibitively time-consuming in early design phases.
LST-based analysis may require the user to interact with the analysis
frequently to check for lost modes and nonphysical results. This
interaction is not only time-consuming, but also requires that the user
have experience interacting with the specific implementation of the
LST-based analysis. Thus, even with powerful computing resources,
LST-based analysis may be impractical when iterating through a large
number of design configurations in the early phases of aircraft design.

[0017] In contrast to those based on LST-based analysis, there are other
transition prediction techniques that require very little user
interaction but may sacrifice accuracy or reliability. Without high
accuracy and reliability, these techniques are less useful for iterating
design configuration in the early phases of aircraft design.

[0018] The techniques described herein can be used to generate a
growth-rate model that reduces or eliminates the need for user
interaction. Further, iteration of the techniques described herein can be
used to provide a prediction of the transition point on a
computer-generated aircraft surface.

SUMMARY

[0019] One exemplary embodiment includes a computer-implemented method of
predicting whether a point on a computer-generated surface is adjacent to
laminar or turbulent fluid flow. A plurality of boundary-layer properties
at the point are obtained from a steady-state solution of a fluid flow in
a region adjacent to the point. A plurality of instability modes are
obtained. Each instability mode is defined by one or more mode
parameters. A vector of regressor weights for the known instability
growth rates in a training dataset is obtained. The vector of regressor
weights is based on the covariance of the known instability growth rates
in the training dataset. For each instability mode in the plurality of
instability modes, a covariance vector is determined. The covariance
vector comprises the covariance of a predicted local growth rate for the
instability mode at the point with the known instability growth rates in
the training dataset. Each covariance vector is used with the vector of
regressor weights to determine a predicted local growth rate for the
instability mode at the point with the boundary-layer properties. Based
on the predicted local growth rates, an n-factor envelope at the point is
determined. The n-factor envelope at the point is indicative of whether
the point is adjacent to laminar or turbulent flow.

DESCRIPTION OF THE FIGURES

[0020] FIG. 1 depicts a supersonic natural-laminar-flow concept jet.

[0021] FIG. 2 depicts a computer-generated simulation of fluid flow over
the surface of a natural-laminar-flow concept jet.

[0022] FIGS. 3a and 3b depict an exemplary fluid flow over a wing.

[0023]FIG. 4 depicts an exemplary process for predicting a point on an
n-factor envelope.

[0024]FIG. 5 depicts the crossflow and streamwise velocity profiles and
the angle between the reference axis and the external streamline.

[0037] The figures depict one embodiment of the present invention for
purposes of illustration only. One skilled in the art will readily
recognize from the following discussion that alternative embodiments of
the structures and methods illustrated herein can be employed without
departing from the principles of the invention described herein.

DETAILED DESCRIPTION

[0038] FIG. 1 illustrates a transonic natural-laminar-flow (NLF) concept
jet. FIG. 2 shows a top view of a portion of a computer-generated
simulation of the NLF jet. The shading on the wing 202 is proportional to
the combined TS and crossflow instabilities on the upper surface at Mach
0.75 at 33,000 feet. The white areas on the wing 202 indicate regions
where turbulent flow is predicted. As seen in FIG. 2, locations near
fuselage 200 exhibit turbulent flow closer to the leading edge 204 of the
wing 202 as compared to locations further away from fuselage 200.

[0039] The results depicted in FIG. 2 are an example of the output of a
computer-generated simulation that allows a designer or engineer to
evaluate the performance of an aircraft surface with respect to laminar
flow. If necessary, changes can be made to the aircraft surface geometry
to optimize or increase the amount of laminar flow. Additional
simulations can be performed for modified aircraft surface geometry and
the results can be compared. To allow for multiple design iterations, it
is advantageous to perform multiple simulations in a short amount of
time. The following process can be used to provide an accurate prediction
of the transition point in a way that reduces simulation time and human
interaction.

[0040] The processes described herein provide for prediction of the
transition from laminar to turbulent flow with reduced amounts of user
interaction. The following discussion provides an example of a simulated
fluid flow over an aircraft surface. However, the processes may also be
applied to a simulated fluid flow over any type of surface subjected to a
fluid flow. For example, the following processes could be applied to the
surface of a space vehicle, land vehicle, watercraft, or other object
having a surface exposed to a fluid flow. In addition, the following
processes can be applied to simulations of various types of fluid flow,
including, for example, a gas fluid flow or liquid fluid flow.

[0041] FIGS. 3a and 3b depict an exemplary aircraft surface, a wing
section 302, and a two-dimensional representation of a fluid flow. The
fluid flow is classified by two regions: outer region 304 and
boundary-layer region 306, 310. As shown in FIGS. 3a and 3b, the laminar
flow portion 306 of the boundary-layer region begins near the leading
edge 326 of the wing surface 302 and is characterized by a sharply
increasing velocity profile 308. Skin friction causes the fluid velocity
close to the wing surface 302 to be essentially zero, with respect to the
surface. The sharply increasing velocity profile 308 develops as the
velocity increases from a near-zero velocity to the boundary-layer edge
velocity.

[0042] The fluid flow within the boundary layer having a velocity profile
308 may be considered laminar because of the layered or sheet-like nature
of the fluid flow. However, the growth of instabilities within the
boundary layer may result in turbulent flow 310 further downstream from
the leading edge. Transition prediction estimates the location on the
surface of the wing where the fluid flow in the boundary layer changes
from laminar to turbulent.

1. Exemplary Process for Transition Prediction

[0043]FIG. 4 depicts a flow chart for an exemplary process 400 for
predicting whether fluid flow near a point of interest (POI) on a
computer-generated aircraft surface is laminar or turbulent. Exemplary
process 400 is suitable for integration into a computer-generated
simulation. Operations described in the flow chart may be repeated on a
point-by-point basis across the computer-generated aircraft surface to
produce an envelope curve. The fluid flow is predicted to transition to
turbulent flow at the location where the envelope, discussed in operation
412 below, exceeds a threshold or critical value. For example, referring
to FIG. 3a, the process may be performed for each surface point 312, 314,
316, and 318 to construct an envelope curve used to determine the
transition point on the wing surface 302.

[0044] The computer-generated aircraft surface may include, for example, a
portion of an airfoil surface or a part of a fuselage surface obtained
from a computer-aided design (CAD) computer software package. In some
cases, the computer-generated aircraft surface includes a surface mesh of
polygons, such as a mesh of triangles that represents the surface of the
aircraft. A fluid-flow mesh may also be defined representing a fluid-flow
region adjacent to the computer-generated aircraft surface. In some
cases, the fluid-flow mesh is generated using, for example, a mesh
generation program or a computation-fluid dynamics (CFD) simulation
module that contains automated mesh generation functionality.

[0045] The POI on the computer-generated aircraft surface where a point on
the envelope curve is to be determined may be selected using the surface
mesh of polygons. For example, the POI may be a vertex of one of the
polygons or a geometrical feature such as a centroid of one of the
polygons. Alternatively, the POI may be an arbitrary point on the
computer-generated aircraft surface that is not associated with any
particular feature of the surface mesh.

[0046] With reference again to FIG. 4, in operation 402 of the process
400, values are determined for boundary-layer properties of the fluid
flow near the POI. Exemplary boundary-layer fluid properties may include
flow velocity, fluid pressure, and temperature. The values of these
properties vary as the POI is chosen to be at different locations on the
computer-generated aircraft surface.

[0047] In some cases, a CFD simulation module can use the surrounding
fluid flow mesh to determine the values of the boundary-layer fluid
properties. In some cases, the results of the CFD simulation module
represent a steady-state solution of the surrounding fluid flow. Values
for the boundary-layer properties relevant to the POI on the
computer-generated aircraft surface are extracted from the steady-state
solution. The boundary-layer properties are selected depending on their
influence in determining whether fluid flow is laminar or turbulent near
the POI.

[0048] In some cases, one or more fluid cells of the fluid-flow mesh are
identified as representing a portion of the boundary-layer fluid flow
near the POI on the computer-generated aircraft surface. Values of
selected boundary-layer properties are extracted from the identified
fluid cells. Exemplary boundary-layer properties that may be relevant to
predicting transition include local Reynolds number, velocity ratios, and
wall-to-external temperature ratios. The relevant boundary-layer
properties may depend, in part, on the type of instability being
analyzed.

[0049] The particular boundary-layer properties that are determined in
operation 402 may depend on the type of instabilities (e.g., TS wave or
crossflow vortex) under consideration. Depending on the type of
instability, different sets of boundary-layer properties may be relevant
to transition prediction. Therefore, each type of instability being
analyzed may require different sets of boundary-layer properties and mode
parameters.

[0050] For example, for TS-wave instabilities with reference to FIG. 5,
relevant boundary-layer properties may include: a Reynolds number defined
by

R=uel/ve,

where

l= {square root over (vex/ue)};

the local Mach number at the boundary-layer edge; five points 504 along
the streamwise velocity profile 502; five points 508 along the crossflow
velocity profile 506; five points along the temperature profile; and the
angle between the reference axis and the external streamline 510.

[0051] In another example, for stationary crossflow instabilities with
reference to FIG. 6, relevant boundary-layer properties may include:

crossflow Reynolds number:

ρ u e δ cf μ ; ##EQU00001##

crossflow velocity ratio:

w max u e ; ##EQU00002##

crossflow shape factor:

y max δ cf ; ##EQU00003##

and the ratio of the wall temperature to external temperature.

[0052] In the examples given above, only four boundary-layer properties
are used when considering stationary crossflow vortices, while twenty
boundary-layer properties are used when considering TS-wave
instabilities.

[0053] In yet another example, the same boundary-layer properties may be
used for both crossflow vortices and TS-wave instabilities. In this
example, the twenty boundary-layer properties discussed above with
respect to TS-wave instabilities may also be used for crossflow vortices.

[0054] While specific examples of boundary-layer properties for particular
types of instabilities are given above, these examples should not be read
to limit the boundary-layer properties that are used. The boundary-layer
properties may be chosen to give the best results.

[0055] With reference again to FIG. 4, in operation 404, a matrix of
parameters defining a plurality of instability modes is constructed. In
an example of operation 404, each mode in the matrix of modes is defined
using at least one of two mode parameters: a temporal frequency and a
spatial spanwise wave number.

[0056] The mode parameters used to define the set of instability modes in
the matrix may depend, in part, on the type of instability being
analyzed. For example, for stationary crossflow vortices, the instability
modes may have a temporal frequency of zero. Thus, the matrix for
crossflow vortices is defined using a range of wave numbers and a single
(zero) temporal frequency. In another example, for TS-wave instabilities,
the instability modes may be defined using both a temporal frequency and
a spatial spanwise wave number. Values for wave numbers and frequency
parameters may be selected at equal intervals across a range of interest.

[0057] In operation 406, a vector of regressor weights of the known
instability growth rates in a training dataset is obtained. In some cases
where the vector of regressor weights has previously been constructed,
operation 406 may be accomplished by loading the vector of regressor
weights from memory.

[0058] If the vector of regressor weights has not been previously
constructed, access to a training dataset is required. The training
dataset includes a plurality of known instability growth rates, each
known instability growth rate having a corresponding input vector. The
known instability growth rates are based on the input vector and
determined using a source that is considered to be accurate. In some
cases, LST-based analysis may be used to determine the known instability
growth rates. LST-based analysis is discussed below with respect to FIG.
15.

[0059] The training input vector includes boundary-layer properties and at
least one mode parameter. The training input vector represents the input
used to calculate the known instability growth rate. Typically, multiple
training input vectors are defined to represent multiple training
instability modes, each training instability mode using the same
boundary-layer property values. Creation of the training dataset is
further discussed below with respect to FIG. 8.

[0060] The vector of regressor weights β may be determined based on a
covariance matrix Σ1 of the known instability growth rates in
the training dataset and a vector of the known instability growth rates
αk in the training dataset according to:

β=Σi-1αk. Equation 1

[0061] Each element of the covariance matrix Σ1 specifies the
covariance of one known instability growth rate in the training dataset
with another known instability growth rate in the training dataset. The
covariance matrix Σ1 is the correlation matrix multiplied by
the variance σ02, which may be determined using the
expected range of variation in growth rates and an optimization technique
(e.g., marginal likelihood) with training data. If the training dataset
includes m known instability growth rates, then the size of the
correlation matrix and the covariance matrix will be m×m. An
element Σij of the covariance matrix Σ1 is the
correlation of the ith known instability growth rate αi
to the jth known instability growth rate αj multiplied by
the variance σ02 as shown in:

Σij=σ02corr(αi,αj).
Equation 2

[0062] In one example of the covariance matrix, the correlation between
two known instability growth rates αi and αj is
based on the distance between the input vectors xi and xj as
shown in:

corr(αi,αj)=r(xi,xj), Equation 3

where r is a correlation function based on the distance between the input
vectors xi and xj.

[0063] In this example, the correlation function r is chosen based on the
assumption that changes in the growth rate are smooth with respect to
changes in the input vector. In other words, the correlation function r
is chosen based on the assumption that the growth rates are infinitely
differentiable with respect to the input vector. A squared exponential
covariance function is an example of one correlation function that is
consistent with this assumption. A squared exponential covariance
function is:

where the input vectors include n elements (boundary-layer properties and
one or more mode parameters), τk is a length-scale parameter for
the kth element of the input vectors, and xi.sup.(k) is the
value of the kth element of the input vector that is associated with
the known instability growth rate αi. The length-scales may be
calculated using an optimization technique such as marginal likelihood
(ML-II maximization) using part of the training dataset where there is
sufficient data. Other covariance functions may be used that make other
assumptions about the relation between the growth rates and the input
vectors.

[0064] Thus, for cases where the vector of regressor weights must be
calculated, equations 1-4 may be used. As long as the training dataset
does not change, the vector of regressor weights will not change.
Accordingly, after calculating the vector of regressor weights, it may be
stored for future use when performing the exemplary process on other
points using the same training dataset.

[0065] Operations 408 and 410 are performed for each instability mode from
the matrix constructed in operation 404. In the discussion of operations
408 and 410 below, the term "current instability mode" refers to one
instability mode of the plurality of instability modes from operation
404.

[0066] In operation 408, a covariance vector is calculated. The covariance
vector comprises the covariance of a predicted local instability growth
rate α0 at the POI with respect to each of the known
instability growth rates in the training dataset. Thus, access to the
training dataset is required for operation 408.

[0067] The covariance vector may be calculated in the same manner as
explained above with respect to operation 406, except using the input
vector x0 for α0 that includes the boundary-layer
properties at the POI from operation 404 and at least one mode parameter
describing the current instability mode. The covariance vector has m
elements.

[0068] In operation 410, the predicted local instability growth rate is
determined using the vector of regressor weights and the covariance
vector Σ2. For example, the relationship:

α0=Σ2β+μ Equation 5

may be used to predict the local instability growth rate at the POI,
where μ is the prior mean, which specifies the local instability
growth rate far away from the input vectors included in the training
dataset. In some cases, the prior mean may be zero.

[0069] Optionally, a confidence measure may be determined for each local
instability growth rate determined in operation 410 based on the variance
of the predicted instability growth rate. The confidence measure may, for
example, be useful in determining whether the training dataset is
suitable for transition prediction on the current computer-generated
aircraft surface. A low confidence measure may indicate that the user
should update the training dataset.

[0070] The variance of the predicted local instability growth rate may be
determined according to

cov(α0)=Σ3Σ1-1Σ3T,
Equation 6

where Σ3 is the covariance vector of the predicted local
instability growth rate with respect to the known instability growth
rates in the training dataset and Σ1 is the covariance matrix
discussed in operation 406 above. The vector Σ3 may be
determined according to the method described above with respect to
equations 2-4.

[0071] In operation 412, a point on the n-factor envelope is determined
based on the local instability growth rates from operation 410 for each
instability mode from operation 404. A point on the n-factor envelope
represents a composite of all the individual n-factors due to the
different instability modes. Generally, the point on the n-factor
envelope is the largest n-factor at the point of all the instability
modes.

[0072] An individual n-factor represents the overall amplification or
attenuation of an instability mode at a particular point. The n-factor at
a point accounts for the cumulative effect of all amplification or
attenuation that occurs prior to that point. In general, instabilities
that become amplified beyond a threshold indicate the presence of
turbulent flow. In some cases, this threshold is called the critical
point.

[0073] An example process for determining a point on the n-factor envelope
includes calculating an n-factor for each instability mode at the POI
using equation 7, below. The n-factor n for a given point x may be
expressed as:

n ( x ) = ∫ x i 0 x - α ( x
' ) x ' , Equation 7 ##EQU00005##

where -a(x') is the predicted local instability growth rate at point x'
as calculated in operation 410 and xi0 is the neutral point, which
is the streamwise point where the instabilities start to grow. To
calculate the n-factor for the POI, it may be necessary to determine
local instability growth rates at other points besides the POI. For
example, local instability growth rates for streamwise points between the
POI and the neutral point may be needed. Operations 404 and 408 discussed
above may be used for each streamwise point to determine the required
local instability growth rates.

[0074] Optionally, a confidence measure may be determined for each of the
individual n-factors determined in operation 412. The confidence measure
may, for example, be useful in determining whether the training dataset
is suitable for transition prediction on the current computer-generated
aircraft surface. A low confidence measure may indicate that the user
should update the training dataset.

[0075] In one case, the confidence measure is determined based on the
variance of the n-factor. The variance may be determined by approximating
the integral in equation 6 by numerical integration as being a weighted
sum of the individual instability growth rates from x0 to x or

i = 1 n - α i c i , ##EQU00006##

where -ai is the predicted instability growth rate at xi,
ci is the weight coefficient for -ai, and there are n predicted
instability growth rates between x0 and x. The variance of the
n-factor may then be determined according to

cov(n(x))=cTΣ4Σ1-1Σ4Tc ,
Equation 8

where c is a weight coefficients vector for the numerical integral above,
Σ4 is the covariance matrix of the n growth rates in the
numerical integral above with respect to the known instability growth
rates in the training dataset, and Σ1 is the covariance matrix
discussed in operation 406 above. The matrix Σ4 may be
determined according to the method described above with respect to
equations 2-4.

[0076] After determining n-factors at the POI for each instability mode
from operation 406, the point on the n-factor envelope can be determined.
The point on the n-factor envelope may be determined by calculating a
pointwise maximum of the individual mode n-factors. However, using the
pointwise maximum for the point on the n-factor envelope may lead to a
non-smooth or irregular n-factor envelope (when viewed as a curve across
the computer-generated aircraft surface). Sometimes a smooth envelope may
be preferred, which may be provided by the following alternative example.
A weighted average a of individual mode n-factors may be calculated as
shown in equation 3, below:

where nk is the n-factor for the individual mode k. As the n-factor
for an individual mode becomes larger compared to the rest of the
n-factors, σ will approach the true maximum

[0077] However, because a is a weighted average, σ will always be a
little less than the true maximum The formula may be modified using the
equivalent of a safety factor. A safety factor may be appropriate
especially when all of the n-factors are small and are about the same
value. Equation 10, below, provides one example for calculating a point
on the n-factor envelope by applying a suitable safety factor to a
weighted average a of the n-factors for the individual modes:

n = σ [ 1 + 0.25 ( 1 - σ 10 ) 2 ] .
Equation 10 ##EQU00008##

[0078] After determining the point on the n-factor envelope with respect
to the POI, the point can be compared to a threshold value or critical
point to determine whether the POI is adjacent to laminar or turbulent
flow. For example, a threshold value or critical point for transition
prediction may be based on empirical data for sufficiently similar
boundary layers. If the point on the n-factor envelope (at the POI) is
less than the threshold value or critical point, then the flow near the
POI is considered laminar. If the point on the n-factor envelope (at the
POI) is greater than this threshold value or critical point, then the
flow at the flow near the POI is considered turbulent.

[0079] The threshold value of the n-factor for the onset of turbulence may
be determined empirically for a given set of conditions. For example, for
aircraft surfaces in wind tunnels, the n-factor critical point for TS
waves may occur at a value ranging from 5 to 9. For aircraft surfaces in
atmospheric flight, the n-factor critical point for TS waves may occur at
a value ranging from 8 to 14.

[0080] The operations of the exemplary process above have been described
with respect to a single POI on the computer-generated aircraft surface
used to determine a point on an n-factor envelope. To determine other
points on the n-factor envelope and construct an n-factor envelope curve,
portions of the above process can be repeated using other points of
interest (POIs) on the computer-generated aircraft surface. For example,
the operations 402, 408, 410, and 412 may be repeated for as many points
as necessary to obtain a satisfactory resolution for the n-factor
envelope across the computer-generated aircraft surface. Exemplary
n-factor envelope curves are shown as profiles in FIGS. 12 and 13 and as
a shaded plot in FIG. 2.

[0081]FIG. 7 depicts a data-flow chart 700 for the exemplary process 400
described above with respect to FIG. 4. The boundary-layer properties 702
obtained from operation 402 and the instability modes 704 obtained from
operation 404 are used as inputs to the data fit module 706 that
represents operations 406, 408, and 410 above. The data fit module 706
outputs the growth rates 708 discussed above in operation 410 for each
instability mode 704 defined by instability mode parameters temporal
frequency f and spanwise wave number 2. As described in operation 412,
based on the local instability growth rates 708 for the instability modes
704, an n-factor 710 is calculated for each of the instability modes 704.
The individual n-factors are then used to create an n-factor envelope
712. The fluid flow can be considered as transitioning from laminar to
turbulent fluid flow at the points or locations closest to the leading
edge on the aircraft surface where the n-factor envelope 712 first
exceeds a threshold value or critical point.

2. Training Dataset Generation

[0082] The flow chart of FIG. 8 depicts an exemplary process 800 for
generating a training dataset. As described above, operation 406 (FIG. 4)
may need access to the training dataset and operation 408 (FIG. 4) does
need access to the training dataset. As briefly discussed above, the
training dataset contains known instability growth rates and an
associated input vector for each known instability growth rate. The input
vector associated with each known instability growth rate represents the
inputs to the analysis used to produce the known local instability growth
rate.

[0083] In operation 802 of the process 800 for generating the training
dataset, the content of the input vectors is defined. The training input
vectors include boundary-layer properties and one or more mode
parameters. As discussed above in conjunction with operation 402 (FIG.
4), the boundary-layer properties used in the input vectors may vary
depending on the type of instability being considered. The same
boundary-layer properties discussed with respect to FIGS. 6 and 7 above,
for TS-waves and stationary crossflow vortices, respectively, may be
selected for inclusion in the training input vectors.

[0084] Like the boundary-layer properties, the relevant mode parameters
may depend on the type of instability being considered. A similar process
for choosing the one or more mode parameters as discussed above with
respect to operation 404 (FIG. 4) may be used to define the one or more
mode parameters included in the training input vectors. For example, both
spanwise wave number and temporal frequency may be important when
considering TS-wave instabilities. Alternatively, when considering
stationary crossflow vortices, the temporal frequency may always be zero
and only the spanwise wave number is needed. Therefore, when considering
TS-wave instabilites, the training input vectors may include both a
spanwise wavenumber and a temporal frequency for the mode, but when
considering stationary crossflow vortices, the training input vectors may
include the spanwise wavenumber without the temporal frequency.

[0085] Boundary-layer properties and mode parameters used in the training
input vectors may also be selected depending on the desired quality of
the dataset. In general, a large training input vector may provide a
training dataset that enables a more robust prediction when used in the
exemplary process. However, larger training input vectors may also
produce a training dataset that is more computationally intensive to use
in the exemplary process.

[0086] In operation 804, a representative set of computer-generated
aircraft surfaces and fluid flows are obtained. For example, aircraft
surfaces with varying characteristics (e.g., wings having different
airfoil profiles or sweep angles) may be selected or defined by the user.
For each combination of a selected computer-generated aircraft surface
and a selected fluid flow, a CFD module or some other suitable means
calculates a steady-state solution.

[0087] In operation 806, boundary-layer properties and a corresponding
boundary-layer solution are determined using each steady-state solution
determined in operation 804. For each steady-state solution, values for
the same set of boundary-layer properties are determined. The
boundary-layer properties may include, for example, temperature, a local
velocity vector, Mach number, Reynolds number, or pressure gradient.

[0088] Optionally in operation 806, boundary-layer properties and
solutions may also be determined based on similarity sequences. This may
be suitable if LST-based analysis is being used to produce growth rates
but may not be suitable if other analysis techniques are being used. A
similarity sequence allows for generation of boundary-layer properties
and solutions by modifying the shape of the boundary layer extracted from
an existing steady-state solution. For example, the boundary-layer
solution determined from a steady-state solution from operation 804 may
be modified to generate a new boundary-layer solution and corresponding
set of boundary-layer properties by warping the boundary-layer profiles
in some advantageous manner This is done without having to perform
additional CFD simulations or empirical analysis and may be particularly
helpful when certain values of the boundary-layer properties are desired
for the training dataset, but it is difficult to find aircraft surfaces
for which operation 804 will produce those desired values. For example, a
similarity sequence can be generated by warping the boundary layer at a
single streamwise station. This may be done by, for example, scaling the
warped boundary-layer profile (e.g., the local velocity profiles 502 and
506 of FIG. 5 and the temperature profile) by the square root of the
distance from the leading edge to fill all streamwise stations with
similar boundary-layer profiles. In this example, the boundary-layer
properties extracted from the new similarity sequence will still cover
the same parameters (e.g., temperature value, local velocity vector
values) but the values for those parameters will be adjusted.

[0089] In operation 808, local instability growth rates are determined.
These growth rates become the known local instability growth rates. In an
example of operation 808, LST-based analysis is performed using the
multiple sets of boundary-layer properties as determined in operation
806. LST-based analysis is described in more detail below with respect to
FIG. 15. For each boundary-layer solution from operation 806, LST-based
analysis is performed for one or more instability modes. As discussed
above, this operation may require that the user interact with LST-based
analysis to check for lost modes and nonphysical results. However, once
the dataset is created, LST-based analysis is not needed to perform the
exemplary process as discussed above with respect to FIG. 4.

[0090] In operation 810, the local growth rates produced in operation 806
are stored in the dataset. In addition, each local growth rate is also
associated with an input vector having contents as defined in operation
802 and includes at least one mode parameter and the boundary-layer
properties that were determined from the same inputs to the analysis in
operation 808 that produced the local instability growth rate.

[0091] Every possible combination of modes and boundary-layer properties
cannot be expressly included in the dataset. The exemplary process as
discussed above with respect to FIG. 4 enables interpolation of the
results in the training dataset, allowing for accurate estimates of
growth rates under conditions not specifically in the training dataset.

[0092] In one example, the training dataset is partitioned and only the
partitions of the training dataset are used in the exemplary process
described above with respect to FIG. 4. This may be useful, for example,
if the dataset is too large to feasibly create a covariance matrix of the
entire dataset. In this case, the dataset may be partitioned and a
covariance matrix and a vector of regressor weights may be constructed
for each partition in accordance with the exemplary process. For example,
with reference to FIG. 9, the training dataset may be partitioned based
on the temporal frequency of the mode associated with the growth rate. A
covariance matrix and a vector of regressor weights may then be
calculated for each partition 902, 904, 906, 908, 910, 912, 914, 916, and
918.

[0093] In another example, all of the data in the training dataset or all
of the data in a particular partition is not needed. In this example,
only a training subset of the training dataset or the partition is used.
Individual datapoints (i.e., known instability growth rates and
associated training input vectors) are added to the training subset until
the exemplary process described above with respect to FIG. 4 can predict
some threshold number of the known instability growth rates that are not
in the training subset to a threshold error tolerance. Optionally, in
adding data to the training subset, priority can be given to those
datapoints with known instability growth rates that the exemplary process
predicts with the highest error using the current training subset.
Addition of datapoints to the training subset may continue until the
training subset meets some error tolerance. For example, individual
datapoints may be added to the training subset until the exemplary
process predicts 90% of the known instability growth rates not in the
training subset with error not exceeding 10% of the known instability
growth rate.

[0094] Operations 802, 804, 806, 808, and 810 may be performed by an end
user, a third-party vendor, or other suitable party. Additionally,
different operations may be performed by different parties. For example,
if an end user does not have experience with LST-based analysis, the end
user may have a third-party vendor perform operation 808 only. In another
example, an end user may have a third-party vendor perform all operations
and supply only the training dataset, the training partitions, or the
training subsets. In yet another example, a third-party vendor may
generate the training dataset but the end user will partition the dataset
or determine what subset of the dataset to use. In still yet another
example, a user may obtain a training dataset from a third-party vendor
and then add additional data to the training dataset to customize it for
the user's needs. This may be useful, for example, if a confidence
measure of the predicted local instability growth rates or the n-factors
according the exemplary process indicates an unacceptable level of error.

3. LST-Based Analysis

[0095]FIG. 15 depicts a data flow for an LST-based analysis used to
predict a transition point. As discussed above, LST-based analysis may be
used in generating the training dataset. FIG. 15 depicts how LST-based
analysis uses boundary-layer solutions 1502, provided by, for example, a
CFD simulation module, to determine local growth rates 1504 of individual
instability modes 1506 in the boundary-layer region of the fluid flow.
LST-based analysis 1508 uses selected mode parameters 1506 (e.g., wave
number λk and frequency fk) and boundary-layer solutions
(e.g., local velocity, and temperature profiles) to compute a streamwise
dimensionless wavelength and a local streamwise amplification factor.
These two quantities are used in a complex-valued eigenvalue analysis
that determines the local instability growth rates 1504 as modeled by a
linear-dynamical system. The type of instability (e.g., TS wave or
crossflow vortex) associated with the local instability growth rate is
determined based on the eigenvector solution corresponding to the
eigenvalue, which is also an output of the LST-based analysis. Thus,
regardless of the type of instability, the LST-based analysis is the
same. The type of instability is determined based on the physical
behavior of the wave.

[0096] LST-based analysis results are generally considered to be accurate
under many conditions. An example of an LST-based analysis tool is the
LASTRAC software tool developed by NASA.

[0097] Using the growth rates 1504, an n-factor 1510 can be determined for
each of the selected modes 1506. Referring to FIG. 15, based on LST-based
analysis results, n-factors 1510 for each instability mode 1506 are
calculated. An n-factor represents the natural logarithm of the ratio of
amplification of an individual instability mode at a given point to its
initial amplification at its neutral point. The n-factor represents the
amplification or attenuation of an instability mode at a given point on
the aircraft surface. As discussed above, if the n-factor reaches a
threshold or critical point, the flow may be considered turbulent.

[0098] An n-factor envelope 1512 is determined using the n-factors from
each selected instability mode 1506. N-factor envelope 1512 represents a
composite of the n-factors for all of the selected instability modes
1506. In some cases, the n-factor envelope 1512 represents the largest
n-factor at a given point for a set of selected instability modes 1506.
For example, the n-factor envelope 1512 may be calculated by taking the
point-wise maximum of the n-factors of the individual instability modes
in the envelope.

4. Results of Computer Experiments

[0099] An exemplary transition prediction process based on the exemplary
process described above was tested using wing surfaces having airfoil
cross sections as shown in FIG. 14. TS-wave instability n-factor results
for individual modes are shown in FIGS. 10 and 11. TS-wave results for
the n-factor envelopes are shown in FIGS. 12 and 13. For the purposes of
these results, the threshold value or critical point is assumed to be 9.

[0100] Computer-generated aircraft surfaces based on airfoil cross
sections shown in FIG. 14 were obtained. The following boundary-layer
conditions were then selected for each aircraft surface:

[0101]
Untapered wings with aspect ratio of 10;

[0102] Leading-edge sweep:
0°, 5°, 15°, 35°;

[0103] Chord Reynolds
numbers: 6, 30, 60 million; and

[0104] Angle of attack: 0°,
5°.

[0105] For each steady-state flow solution, LST-based analysis was used to
determine local instability growth rates for individual modes.
Additionally, LST-based analysis was used with similarity sequences to
produce additional local instability growth rates. Because TS-wave
instabilities were being considered, the growth rates were then stored in
a dataset with the wave number, the mode frequency, the Reynolds number,
the local Mach number, five points along the streamwise velocity profile,
five points along the crossflow velocity profile, five points along the
temperature profile, and the angle between the reference axis and the
external streamline Initially, a dataset of about 300,000 known
instability growth rates with associated input vectors was generated.

[0106] The dataset was partitioned based on mode frequency. The exemplary
process described above with respect to FIG. 4 was performed using a
subset of the partitions.

[0107] FIGS. 10 and 11 are graphs showing a comparison among n-factors for
individual instability modes calculated with the exemplary process
described above with respect to FIG. 4 and LST-based analysis. FIGS. 12
and 13 are graphs of n-factor envelope results according to an envelope
modeling technique, n-factor envelope results according to the exemplary
transition prediction process, and individual instability mode n-factor
results as calculated by LST-based analysis. As can be seen, the n-factor
envelope calculated based on the exemplary transition prediction
technique better predicts the results of the LST-based analysis as
compared to the envelope modeling technique.

[0109] Processor 1602 is a computer processor capable of receiving and
executing computer-executable instructions for performing any of the
processes described above. Computer system 1600 may include more than one
processor for performing the processes. The computer-executable
instructions may be stored on one or more types of nontransitory storage
media including RAM 1610, hard drive storage 1612, or other
computer-readable storage media 1614. Other computer-readable storage
media 1614 include, for example, CD-ROM, DVD, magnetic tape storage,
magnetic disk storage, solid-state storage, and the like.

[0110]FIG. 17 depicts an exemplary computer network for distributing the
processes described above to multiple computers at remote locations. One
or more servers 1710 may be used to perform portions of the process
described above. For example, one or more servers 1710 may store and
execute computer-executable instructions for receiving information for
generating a computer-generated simulation. The one or more servers 1710
are specially adapted computer systems that are able to receive input
from multiple users in accordance with a web-based interface. The one or
more servers 1710 are able to communicate directly with one another using
a computer network 1720, including a local area network (LAN) or a wide
area network (WAN), such as the Internet.

[0111] One or more client computer systems 1740 provide an interface to
one or more system users. The client computer systems 1740 are capable of
communicating with the one or more servers 1710 over the computer network
1720. In some embodiments, the client computer systems 1740 are capable
of running a Web browser that interfaces with a Web-enabled system
running on one or more server machines 1710. The Web browser is used for
accepting input data from the user and presenting a display to the user
in accordance with the exemplary user interface described above. The
client computer 1740 includes a computer monitor or other display device
for presenting information to the user. Typically, the client computer
1740 is a computer system in accordance with the computer system 1600
depicted in FIG. 16.

[0112] Although the invention has been described in considerable detail
with reference to certain embodiments thereof, other embodiments are
possible, as will be understood by those skilled in the art.